The Brewers backed up Greinke with an 11-hit attack, including a three-run homer by Taylor Green that finished the scoring in the eighth.
Milwaukee took the lead with a four-run fourth inning, during which Brooks Conrad contributed a two-run single.
The loss was charged to Paul Maholm (4-5), who surrendered four runs on six hits in four innings.
Chicago remained stuck on 19 wins this season, tied with San Diego for fewest in the majors.
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